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Questions and Answers
What does the Remainder Theorem state?
What does the Remainder Theorem state?
- A polynomial f(x) has a factor (x-k) if and only if f(x)=0
- Complex imaginary roots must work in teams of two
- The remainder of a polynomial divided by x-a is p(a) (correct)
- A polynomial of degree n has n roots
What is the Factor Theorem?
What is the Factor Theorem?
- A function must have at least one zero
- Every rational root of the polynomial can be written in the form of p/q
- A polynomial of degree n has n roots
- If a polynomial f(x) has a factor (x-k) if and only if f(x)=0 (correct)
What does the Rational Root Theorem indicate?
What does the Rational Root Theorem indicate?
Every rational root of the polynomial can be written in the form of p/q where p is a factor of the constant term and q is a factor of the leading coefficient.
What does the Irrational Root Theorem state?
What does the Irrational Root Theorem state?
A polynomial of degree n has n roots.
A polynomial of degree n has n roots.
According to the Fundamental Theorem of Algebra, a polynomial function must have at least one zero.
According to the Fundamental Theorem of Algebra, a polynomial function must have at least one zero.
What does the Complex Conjugate Theorem state?
What does the Complex Conjugate Theorem state?
What does the Location Principle describe?
What does the Location Principle describe?
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Study Notes
Remainder Theorem
- When dividing a polynomial function p(x) by x-a, the remainder r is found by evaluating p(a).
Factor Theorem
- A polynomial f(x) contains a factor (x-k) if and only if f(k) equals zero. This signifies k is a root of the polynomial.
Rational Root Theorem
- For a polynomial p(x) with integer coefficients, any rational root can be expressed as p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
Irrational Root Theorem
- In a polynomial p(x) with rational coefficients, if a root takes the form a+b√c (where a and b are rational, and √c is irrational), then its conjugate a-b√c is also a root.
Number of Roots Theorem
- A polynomial of degree n possesses exactly n roots, considering both real and complex roots.
Fundamental Theorem of Algebra
- Every polynomial function has at least one complex zero, which may be real or non-real.
Complex Conjugate Theorem
- If a polynomial contains complex roots, they occur in conjugate pairs; if 2-i is a root, then its conjugate, 2+i, must also be a root.
Location Principle
- If a polynomial function p takes on opposite signs at two points, p(x1) and p(x2), then there exists a real number r between x1 and x2 where p(r) equals zero, confirming that r is a root.
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